6Systems of particles

IA Dynamics and Relativity

6.3 The two-body problem

The two-body problem is to determine the motion of two bodies interacting only

via gravitational forces.

The center of mass is at

R =

1

M

(m

1

r

1

+ m

2

r

2

),

where M = m

1

+ m

2

.

The magic trick to solving the two-body problem is to define the separation

vector (or relative position vector)

r = r

1

− r

2

.

Then we write everything in terms of R and r.

r

1

= R +

m

2

M

r, r

2

= R −

m

1

M

r.

r

2

r

1

R

r

Since the external force

F

=

0

, we have

¨

R

=

0

, i.e. the center of mass moves

uniformly.

Meanwhile,

¨

r =

¨

r

1

−

¨

r

2

=

1

m

1

F

12

−

1

m

2

F

21

=

1

m

1

+

1

m

2

F

12

We can write this as

µ

¨

r = F

12

(r),

where

µ =

m

1

m

2

m

1

+ m

2

is the reduced mass. This is the same as the equation of motion for one particle of

mass

µ

with position vector

r

relative to a fixed origin — as studied previously.

For example, with gravity,

µ

¨

r = −

Gm

1

m

2

ˆ

r

|r|

2

.

So

¨

r = −

GM

ˆ

r

|r|

2

.

For example, give a planet orbiting the Sun, both the planet and the sun moves

in ellipses about their center of mass. The orbital period depends on the total

mass.

It can be shown that

L = MR ×

˙

R + µr ×

˙

r

T =

1

2

M|

˙

R|

2

+

1

2

µ|

˙

r|

2

by expressing r

1

and r

2

in terms of r and R.